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Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 150145, 20 pages
doi:10.1155/2012/150145
Research Article
Viscosity Approximation Methods for
Equilibrium Problems, Variational Inequality
Problems of Infinitely Strict Pseudocontractions in
Hilbert Spaces
Aihong Wang
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Aihong Wang, [email protected]
Received 23 April 2012; Accepted 2 August 2012
Academic Editor: Hong-Kun Xu
Copyright q 2012 Aihong Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We introduce an iterative scheme by the viscosity approximation method for finding a common
element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely
strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces.
Our results improve and extend the corresponding results announced by many others recently.
1. Introduction
Let H be a real Hilbert space and let C be a nonempty convex subset of H.
A mapping S of C is said to be a κ-strict pseudocontraction if there exists a constant
κ ∈ 0, 1 such that
Sx − Sy2 ≤ x − y2 κI − Sx − I − Sy2 ,
1.1
for all x, y ∈ C; see 1. We denote the set of fixed points S by FS i.e., FS {x ∈ C : Sx x}.
Note that the class of strict pseudocontraction strictly includes the class of nonexpansive mappings which are mappings S on C such that
Sx − Sy ≤ x − y,
1.2
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Journal of Applied Mathematics
for all x, y ∈ C. That is, S is nonexpansive if and only if S is a 0-strict pseudocontraction.
Let Φ be a bifunction from C × C to R, where R is the set of real numbers. The equilibrium
problem for Φ : C × C → R is to find x ∈ C such that
Φ x, y ≥ 0,
∀y ∈ C.
1.3
The set of solutions of 1.3 is denoted by EP Φ. Given a mapping B : C → H, let Φx, y Bx, y − x for all x, y ∈ C. Then the classical variational inequality problem is to find x ∈ C
such that Bx, y − x ≥ 0. We denote the solution of the variational inequality by VIC, B;
that is
VIC, B x ∈ C : Bx, y − x ≥ 0 .
1.4
Let A be a strongly positive linear-bounded operator on H if there is a constant γ > 0 with
property
Ax, x ≥ γx2 ,
1.5
∀x ∈ H.
A typical problem is to minimize a quadratic function over the set of the fixed points a
nonexpansive mapping on a real Hilbert space H:
1
min Ax, x − x, b,
x∈E 2
1.6
where A is a linear-bounded operator, E is the fixed point set of a nonexpansive mapping
S on H, and b is a given point in H. The problem 1.3 is very general in the sense that it
includes, as special cases, optimization problems, variational inequalities, minimax problems,
the Nash equilibrium problem in noncooperative games, and others; see 1–11. In particular,
Combettes and Hirstoaga 4 proposed several methods for solving the equilibrium problem.
On the other hand, Mann 6, Shimoji and Takahashi 8 considered iterative schemes for
finding a fixed point of a nonexpansive mapping. Further, Acedo and Xu 12 projected new
iterative methods for finding a fixed point of strict pseudocontractions.
In 2006, Marino and Xu 7 introduced the general iterative method: for x1 x ∈ C,
xn1 αn γfxn I − αn AT xn ,
n ≥ 1.
1.7
They proved that the sequence {xn } of parameters satisfies appropriate condition and that the
sequence {xn } generated by 1.7 converges strongly to the unique solution of the variational
inequality γf − Aq, p − q ≤ 0, p ∈ FT . Recently, Liu 5 considered a general iterative
method for equilibrium problems and strict pseudocontractions:
1
y − un , un − xn ≥ 0,
Φ un , y rn
yn βn un 1 − βn Sun ,
xn1 εn γfxn I − εn Aun ,
∀y ∈ C,
1.8
∀n ≥ 1,
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where S is a k-strict pseudocondition mapping and {εn }, {βn } are sequences in 0, 1. They
proved that under certain appropriate conditions over {εn }, {βn }, and {rn }, the sequences
{xn } and {un } both converge strongly to some q ∈ FS EP Φ, which solves some
variational inequality problems. Tian 10 proposed a new general iterative algorithm: for
nonexpansive mapping T : H → H with FT / φ,
xn1 αn γfxn I − μαn F T xn ,
∀n ≥ 1,
1.9
where F is a k-Lipschitzian and η-strong monotone operator. He obtained that the sequence
xn generated by 1.9 converges to a point q in FT , which is the unique solution of the
variational inequality γf − Aq, p − q ≤ 0, p ∈ FT . Very recently, Wang 13 considered a
general composite iterative method for infinite family strict pseudocontractions: for x1 x ∈
C,
xn1
yn βn xn 1 − βn Wn xn ,
αn γfxn I − μαn F yn , ∀n ≥ 1,
1.10
where Wn is a mapping defined by 2.5, F is a k-Lipschitzian, and η-strongly monotone
operator. With some appropriate condition, the sequence {xn } generated by 1.10 converges
strongly to a common element of the fixed point of an infinite family of λi -strictly
pseudocontractive mapping, which is a unique solution of the variational inequality γf −
Aq, p − q ≤ 0, p ∈ FT . Kumam proposed many algorithms for the equilibrium and
the fixed point problems with k-strict pseudoconditions; see 14–16. In particular, in 2011,
Kumam and Jaiboon 14 considered a system of mixed equilibrium problems, variational
inequality problems, and strict pseudocontractive mappings:
x1 ∈ E,
un ∈ E,
vn ∈ E,
φ ,ϕ
un Tr 1 xn ,
φ ,ϕ
vn Ts 2 xn ,
zn PE un − μn Cun ,
1.11
yn PE vn − λn Cvn ,
kn an Sk xn bn yn cn zn ,
xn1 εn γfxn βn xn 1 − βn I − εn A kn ,
∀n ≥ 1,
where S is a k-strict pseudocondition mapping. They proved that under certain appropriate
conditions over {εn }, {βn }, {rn }, {an }, {bn }, {cn }, {λn }, and {μn }, the sequence {xn } converges
strongly to a point q ∈ Θ which is the unique solution of the variational inequality A −
γfq, x − q ≥ 0. Inprasit 17 proposed a viscosity approximation methods to solving the
generalized equilibrium and fixed point problems of finite family of nonexpansive mapping
in Hilbert spaces.
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In this paper, motivated by the above facts, we use the viscosity approximation method
to find a common element of the set of solutions of the equilibrium problem VIC, B and the
set of fixed points of a infinite family of strict pseudocontractions.
2. Preliminaries
Throughout this paper, we always write for weak convergence and → for strong
convergence. We need some facts and tools in a real Hilbert space H which are listed as
below.
Lemma 2.1. Let H be a real Hilbert space. There hold the following identities:
i x − y2 x2 − y2 − 2x − y, y, ∀x, y ∈ H,
ii tx 1 − ty2 tx2 1 − ty2 − t1 − tx − y2 , ∀t ∈ 0, 1, ∀x, y ∈ H.
Lemma 2.2 see 18. Assume that {αn } is a sequence of nonnegative real numbers such that
αn1 ≤ 1 − ρn αn σn ,
2.1
where {ρn } is a sequence in 0, 1 and {σn } is a sequence such that
i
∞
n1
ρn ∞,
ii lim supn → ∞ σn /ρn ≤ 0 or
∞
n1
|σn | < ∞.
Then limn → ∞ an 0.
Recall that given a nonempty closed convex subset C of a real Hilbert space H, for any
x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that
x − PC x ≤ x − y,
2.2
for all y ∈ C. Such a PC is called the metric or the nearest point projection of H onto C. As
we all know, y PC x if and only if there holds the relation:
x − y, y − z ≥ 0
∀z ∈ C.
2.3
Lemma 2.3 see 13. Let A : H → H be an L-Lipschitzian and η-strongly monotone operator on
a Hilbert space H with L > 0, η > 0, 0 < μ < 2η/L2 , and 0 < t < 1. Then S I − tμA : H → H
is a contraction with contractive coefficient 1 − tτ and τ 1/2μ2η − μL2 .
Lemma 2.4 see 1. Let S : C → C be a κ-strict pseudocontraction. Define T : C → C by
T x λx 1 − λSx for each x ∈ C. Then, as λ ∈ κ, 1, T is a nonexpansive mapping such that
FT FS.
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5
Lemma 2.5 see 10. Let H be a Hilbert space and f : H → H a contraction with coefficient
0 < α < 1, and A : H → H an L-Lipschitzian continuous operator and η-strongly monotone with
L > 0, η > 0. Then for 0 < γ < μη/α,
2
x − y, μA − γf x − μA − γf y ≥ μη − γα x − y ,
x, y ∈ H.
2.4
That is, μA − γf is strongly monotone with coefficient μη − γα.
Let {Sn } be a sequence of κn -strict pseudo-concontractions. Define Sn θn I 1 −
θn Sn , θn ∈ κn , 1. Then, by Lemma 2.4, Sn is nonexpansive. In this paper, we consider the
mapping Wn defined by
Un,n1 I,
Un,n tn Sn Un,n1 1 − tn I,
Un,n−1 tn−1 Sn−1 Un,n 1 − tn−1 I,
···,
2.5
Un,i ti Si Un,i1 1 − ti I,
···,
Un,2 t2 S2 Un,3 1 − t2 I,
Wn Un,1 t1 S1 Un,2 1 − t1 I.
Lemma 2.6 see 8. Let C be a nonempty closed convex subset of a strictly convex Banach space E,
∅, and let t1 , t2 , . . . be
let S1 , S2 , . . . be nonexpansive mappings of C into itself such that ∩∞
i1 FSi /
real numbers such that 0 < ti ≤ b < 1, for every i 1, 2, . . .. Then, for any x ∈ C and k ∈ N, the limit
limn → ∞ Un,k x exists.
Using Lemma 2.6, one can define the mapping W of C into itself as follows:
Wx : lim Wn x lim Un,1 x,
n→∞
n→∞
x ∈ C.
2.6
Lemma 2.7 see 8. Let C be a nonempty closed convex subset of a strictly convex Banach space E.
Let S1 , S2 , . . . be nonexpansive mappings of C into itself such that ∩∞
∅, and let t1 , t2 , . . . be
i1 FSi /
real numbers such that 0 < ti ≤ b < 1, for all i ≥ 1. If K is any bounded subset of C, then
lim supWx − Wn x 0.
n → ∞ x∈K
2.7
Lemma 2.8 see 3. Let C be a nonempty closed convex subset of a Hilbert space H, let {Si : C →
∅, and let t1 , t2 , . . . be real numbers
C} be a family of infinite nonexpansive mappings with ∩∞
i1 FSi /
such that 0 < ti ≤ b < 1, for every i 1, 2, . . .. Then FW ∩∞
i1 FSi .
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Journal of Applied Mathematics
For solving the equilibrium problem, let us assume that the bifunction Φ satisfies the
following conditions:
A1 Φx, x 0 for all x ∈ C;
A2 Φ is monotone; that is Φx, y Φy, x ≤ 0 for any x, y ∈ C;
A3 for each x, y, z ∈ C, lim supt → 0 Φtz 1 − tx, y ≤ Φx, y;
A4 Φx, · is convex and lower semicontinuous for each x ∈ C.
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.9 see 2. Let C be a nonempty closed convex subset of H, let Φ be bifunction from C × C
to R satisfying (A1)–(A4), and let r > 0 and x ∈ H. Then there exists z ∈ C such that
1
Φ z, y y − z, z − x ≥ 0,
r
2.8
∀y ∈ C.
Lemma 2.10 see 4. Let φ be a bifunction from C × C into R satisfying (A1)–(A4). Then, for any
r > 0 and x ∈ H, there exists z ∈ C such that
1
Φ z, y y − z, z − x ≥ 0,
r
2.9
∀y ∈ C.
Further, if Tr x {z ∈ C; Φz, y 1/ry − z, z − x ≥ 0, ∀y ∈ C}, then the following hold:
1 Tr is single-valued;
2 Tr is firmly nonexpansive;
3 FTr EP φ;
4 EP φ is closed and convex.
Lemma 2.11 see 9. Let {xn } and {zn } be bounded sequences in a Banach space, and let {βn } be
a sequence of real numbers such that 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1 for all n 0, 1, 2, . . ..
Suppose that xn1 1 − βn zn βn xn for all n 0, 1, 2, . . .. and lim supn → ∞ zn1 − zn − xn1 −
xn ≤ 0. Then limn → ∞ zn − xn 0.
Lemma 2.12 see 11. Let C, H, F, and Tr x be as in Lemma 2.9. Then the following holds:
Ts x − Tt x2 ≤ Ts x − Tt x, Ts x − x,
2.10
for all s, t > 0, and x ∈ H.
Lemma 2.13 see 13. Let H be a Hilbert space, and let C be a nonempty closed convex subset
of H, and T : C → C a nonexpansive mapping with FT / ∅. If {xn } is a sequence in C weakly
converging to x and if {I − T xn } converges strongly to y, then I − T x y.
3. Main Results
Now we start and prove our main result of this paper.
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Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ be
a bifunction from C × C → R satisfying (A1)–(A4). Let Si : C → C be a family κi -strict
∅. Let f be
pseudocontractions for some 0 ≤ κi < 1. Assume the set Ω VIC, B∩∞
i1 FSi ∩ EP φ /
a contraction of H into itself with α ∈ 0, 1, and let A be a strongly positive linear bounded operator
on H with coefficient γ > 0 and 0 < γ < γ/γ. Let B : C → H be an ξ-inverse strongly monotone
mapping. Let Wn be the mapping generated by Si and ti as in 2.5. Let {xn } be a sequence generated
by the following algorithm:
1
φ zn , y y − zn , zn − xn ≥ 0,
λn
yn PC I − μn B zn ,
xn1
3.1
Kn αn xn 1 − αn Wn yn ,
εn γfxn βn xn 1 − βn I − εn A Kn ,
where {εn }, {βn }, {αn }, and {λn } are sequences in 0, 1. Assume that the control sequences satisfy
the following restrictions:
i limn → ∞ εn 0 and Σ∞
n1 εn ∞;
ii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1;
iii 0 < limn → ∞ λn /λn1 1;
iv limn → ∞ |αn1 − αn | 0;
v 0 < μn ≤ 2ξ;
vi limn → ∞ αn a.
Then {xn } converges strongly to q ∈ Ω which is the unique solution of the variational inequality
A − γf q, x − q ≥ 0,
∀x ∈ Ω,
3.2
or equivalent q PΩ I − A γfq, where P is a metric projection mapping form H onto Ω.
Proof. Since εn → 0, as n → ∞, we may assume, without loss of generality, that εn ≤ 1 −
βn A−1 for all n ∈ N. By Lemma 2.3, we know that if 0 ≤ ρ ≤ A−1 , then I − ρA ≤ 1 − ργ.
We will assume that I − A ≤ 1 − γ. Since A is a strongly positive bounded linear operator
on H, we have
A sup{|Ax, x| : x ⊆ H, x 1}.
3.3
1 − βn I − εn A x, x 1 − βn x2 − εn Ax, x
≥ 1 − βn x2 − εn A
3.4
Observe that
≥ 0.
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So this shows that 1 − βn I − εn A is positive. It follows that
1 − βn I − εn A sup 1 − βn I − εn A x, x : x ⊆ H, x 1
sup 1 − βn − εn Ax, xx ⊆ H, x 1 ≤ 1 − βn − εn γ.
3.5
Step 1. We claim that the mapping PΩ I − A γf where Ω ∞
EP Φ has a
i1 FSi unique fixed point. Let f be a contraction of H into itself with α ∈ 0, 1. Then, we have
PC I − A γf x − PC I − A γf y ≤ I − A γf x − I − A γf y ≤ I − Ax − y γ fx − f y ≤ 1 − γ x − y γαx − y
1 − γ − γα x − y,
3.6
for all x, y ∈ H. Since 0 < 1 − γ − γα < 1, it follows that PΩ I − A γf is a contraction of
H into itself. Therefore the Banach contraction mapping principle implies that there exists a
unique element q ∈ H such that q PΩ I − A γfq.
Step 2. We shall show that I − μn B is nonexpansive. Let x, y ∈ C. Since B is ξ-inverse
strongly monotone and λn < 2α for all n ∈ N, we obtain
I − μn B x − I − μn B y2 x − y − μn Bx − By 2
2
x − y − 2μn x − y, Bx − By
2
μ2n Bx − By
2
2
2
≤ x − y − 2ξμn Bx − By μ2n Bx − By
2
2 2
x − y μn μn − 2ξ Bx − By ≤ x − y ,
3.7
where μn ≤ 2ξ, for all n ∈ N. So we have that the mapping I − λn A is nonexpansive.
Step 3. We claim that {xn } is bounded.
Let p ∈ Ω; from Lemma 2.10, we have
p PC p − μn Bp Tλn p,
zn − p Tλ xn − Tλ p
n
n
≤ xn − p.
3.8
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9
Note that
yn − p PC I − μn B zn − p
PC I − μn B zn − PC I − μn B p
≤ zn − μn Bzn − p − μn Bp I − μn B zn − p ≤ zn − p
≤ xn − p,
Kn − p αn xn 1 − αn Wn yn − p
αn xn − p 1 − αn Wn yn − p ≤ αn xn − p 1 − αn Wn yn − p
≤ αn xn − p 1 − αn yn − p
≤ xn − p.
3.9
It follows that
xn1 − p εn γfxn βn xn 1 − βn I − εn A Kn − p
εn γfxn − Ap βn xn − p 1 − βn I − εn A Kn − p ≤ εn γfxn − Ap βn xn − p 1 − βn − εn γ Kn − p
≤ εn γfxn − Ap βn xn − p 1 − βn − εn γ xn − p
≤ 1 − εn γ xn − p εn γ fxn − f p εn γf p − Ap
≤ 1 − εn γ xn − p εn γαxn − p εn γf p − Ap
γf p − Ap
1 − γ − αγ εn xn − p γ − αγ εn
γ − αγ
γf p − Ap
.
≤ max xn − p,
γ − αγ
3.10
By simple induction, we have
γf p − Ap
xn − p ≤ max x1 − p,
,
γ − αγ
∀n ∈ N.
Hence {xn } is bounded. This implies that {Kn }, {fxn } are also bounded.
3.11
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Step 4. Show that lim supn → ∞ xn1 − xn 0.
Observing that zn Tλn xn and zn1 Tλn1 , we get
zn1 − zn Tλn1 xn1 − Tλn xn Tλn1 xn1 − Tλn1 xn Tλn1 xn − Tλn xn 3.12
≤ xn1 − xn Tλn1 xn − Tλn xn .
By Lemma 2.10, we obtain
1
Φ Tλn xn , y y − Tλn xn , Tλn un − xn ≥ 0,
λn
∀y ∈ C,
1 y − Tλn1 xn , Tλn1 xn − xn ≥ 0,
Φ Tλn1 xn , y λn1
3.13
∀y ∈ C.
In particular, we have
ΦTλn xn , Tλn1 xn ΦTλn1 xn , Tλn xn 1
Tλn1 xn − Tλn xn , Tλn xn − xn ≥ 0,
λn
1
λn1
3.14
Tλn xn − Tλn1 xn , Tλn1 xn − xn ≥ 0.
Summing up 3.14 and using A2 , we obtain
1
λn1
Tλn xn − Tλn1 xn , Tλn1 xn − xn 1
Tλn1 xn − Tλn xn , Tλn xn − xn ≥ 0,
λn
3.15
for all y ∈ C. It follows that
Tλ xn − xn Tλn xn − xn
−
Tλn xn − Tλn1 xn , n1
≥ 0.
λn1
λn
3.16
This implies
λn
Tλn1 xn − xn λn1
λn
Tλn1 xn − Tλn xn , Tλn xn − Tλn1 xn 1 −
Tλn1 xn − xn .
λn1
3.17
λn Tλ xn − Tλn xn Tλn1 xn xn .
Tλn1 xn − Tλn xn ≤ 1 −
λ n1
3.18
0≤
Tλn1 xn − Tλn xn , Tλn xn − xn −
It follows that
2
n1
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11
Hence, we obtain
λn L,
Tλn1 xn − Tλn xn ≤ 1 −
λn1 2
3.19
where L sup{xn Tλn1 xn : n ∈ N}. By 3.12 and 3.19, we obtain
zn1 − zn ≤ un1 − un Tλn1 un − Tλn un λn L,
≤ xn1 − xn 1 −
λn1 yn1 − yn PC I − μn B zn1 − PC I − μn B zn ≤ I − μn B zn1 − zn 3.20
λn L.
≤ zn1 − zn ≤ xn1 − xn 1 −
λn1 From 2.5, we have
Wn1 yn − Wn yn t1 S Un1,2 yn − t1 S Un,2 yn 1
1
≤ t1 Un1,2 yn − Un,2 ≤ t1 t2 S2 Un1,3 yn − t2 S2 Un,3 yn ≤ t1 t2 Un1,3 yn − Un,3 yn ≤ ···
≤ M1
n
ti ,
i1
where M1 supn {Un1,n1 yn − Un,n1 yn }.
Note that
Kn1 − Kn αn1 xn1 1 − αn1 Wn1 yn1 − αn xn − 1 − αn Wn yn αn1 xn1 − xn αn1 xn 1 − αn1 Wn1 yn1 − Wn yn
−αn xn 1 − αn1 Wn yn − 1 − αn Wn yn αn1 xn1 − xn αn1 − αn xn 1 − αn1 Wn1 yn1 − Wn yn
αn − αn1 Wn yn 3.21
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Journal of Applied Mathematics
≤ αn1 xn1 − xn |αn1 − αn |xn 1 − αn1 Wn1 yn1 − Wn yn |αn − αn1 |Wn yn ≤ αn1 xn1 − xn |αn1 − αn |xn 1 − αn1 Wn1 yn1 − Wn1 yn Wn1 yn − Wn yn |αn − αn1 |Wn yn ≤ αn1 xn1 − xn |αn1 − αn |xn n
λn L M1
1 − αn1 xn1 − xn 1 − αn1 1 −
ti
λn1 i1
|αn − αn1 |Wn yn λn L ,
1 − αn1 M1
ti 1 −
λn1 i1
≤ xn1 − xn 2|αn1 −
αn |M2
n
3.22
where M2 sup{xn , Wn yn }.
Suppose xn1 βn xn 1 − βn ln , then ln xn1 − βn xn /1 − βn εn γfxn 1 −
βn I − εn FKn /1 − βn .
Hence, we have
ln1 − ln εn1 γfxn1 1 − βn1 I − εn A Kn1 εn γfxn 1 − βn I − εn A Kn
−
1 − βn1
1 − βn
εn1
εn
εn1
εn
γfxn1 −
γfxn Kn1 − Kn AKn −
AKn1
1 − βn1
1 − βn
1 − βn
1 − βn1
εn1 εn γfxn1 − AKn1 AKn − γfxn Kn1 − Kn .
1 − βn1
1 − βn
3.23
Then
ln1 − ln − xn1 − xn ≤
εn1 γfxn1 AKn1 1 − βn1
≤
εn AKn γfxn Kn1 − Kn − xn1 − xn 1 − βn
εn1 γfxn1 AKn1 εn AKn γfxn 1 − βn1
1 − βn
n
λn L M1
2|αn1 − αn |M2 1 − αn1 1 −
ti .
λn1 i1
3.24
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13
Combining with i, iii, and iv, we have
lim supln1 − ln − xn1 − xn ≤ 0.
n→∞
3.25
Hence, by Lemma 2.11, we obtain ln − xn → 0 as n → ∞. It follows that
lim xn1 − xn lim 1 − βn ln − xn 0.
n→∞
n→∞
3.26
We also know that
xn1 − xn εn γfxn βn xn 1 − βn I − εn A Kn − xn
εn γfxn − Axn 1 − βn I − εn A Kn − xn .
3.27
lim Kn − xn 0.
3.28
So
n→∞
Step 5. We claim that xn − Wxn → 0.
Observe that
xn − Wn yn xn − xn1 xn1 − yn Kn − Wn yn ≤ xn − xn1 xn1 − yn αn xn − Wn yn .
3.29
From A1, A3, and 3.28, using step 2, we have
1 − αn xn − Wn yn ≤ xn − xn1 εn γfxn βn xn − βn Kn − εn AKn ≤ xn − xn1 εn γfxn − AKn βn xn − Kn .
3.30
This implies that
xn − Wn yn −→ 0 as n −→ ∞.
3.31
Next we want to show limn → ∞ xn − yn 0.
EP Φ; we have
Let p ∈ ∞
i1 FSi zn − p2 Tλ xn − Tλ p2
n
n
≤ Tλn xn − Tλn p, xn − p
zn − p, xn − p
1 zn − p2 xn − p2 − xn − zn 2 .
2
3.32
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Journal of Applied Mathematics
Therefore
zn − p2 ≤ xn − p2 − xn − zn 2 .
3.33
Note that
Kn − p αn xn 1 − αn Wn yn − p2
2
αn xn − p 1 − αn Wn yn − p 2
2
≤ αn xn − p 1 − αn yn − p
2
2
≤ αn xn − p 1 − αn zn − p
3.34
2
2
≤ αn xn − p 1 − αn xn − p − xn − zn 2
2
xn − p − 1 − αn xn − zn 2 .
From 3.34, we have
xn1 − p2 εn γfxn βn xn 1 − βn I − εn A Kn − p2
2 2
2
εn γfxn − Ap βn xn − p 1 − βn I − εn A Kn − p
2 2
2
≤ εn γfxn − Ap βn xn − p 1 − βn − εn γ Kn − p
2
2
≤ εn γfxn − Ap βn xn − p
2
1 − βn − εn γ xn − p − 1 − αn xn − zn 2
3.35
2 2 εn γfxn − Ap 1 − εn γ xn − p − 1 − βn − εn γ 1 − αn ,
2 2 xn − zn 2 ≤ εn γfxn − Ap xn − p 1 − βn − εn γ 1 − αn xn − zn 2 .
It follows that
2 2 2
1 − βn − εn γ 1 − αn xn − zn 2 ≤ εn γfxn − Ap xn − p − xn1 − p
2 εn γfxn − Ap xn − p xn1 − p
3.36
× xn − xn1 .
From conditions i, vi and 3.26, we have
xn − zn −→ 0.
3.37
Journal of Applied Mathematics
15
We also compute
yn − p2 PC I − μn B zn − PC I − μn B p
2
≤ zn − μn Bzn − p − μn Bp 2
2
zn − p − 2μn zn − p, Bzn − Bp μ2n Bzn − Bp
2
2
≤ xn − p − μn 2ξ − μn Bzn − Bp ,
Kn − p2 αn xn 1 − αn Wn yn − p2
2
αn xn − p − 1 − αn Wn yn − p 2
2
≤ αn xn − p 1 − αn yn − p
2
2
2 αn xn − p 1 − αn xn − p − μn 2ξ − μn Bzn − Bp
3.38
3.39
2
2
xn − p − μn 1 − αn 2ξ − μn Bzn − Bp .
So, from 3.39, we get
xn1 − p2 εn γfxn βn xn 1 − βn I − εn A Kn − p2
2
2
εn γfxn − Ap βn xn − p 1 − βn I − εn A Kn − p
2 2
2
≤ εn γfxn − Ap βn xn − p 1 − βn − εn γ Kn − p
2 2
≤ εn γfxn − Ap βn xn − p 1 − βn − εn γ
2
2 × xn − p − μn 1 − αn 2ξ − μn Bzn − Bp
2
2 ≤ εn γfxn − Ap xn − p
2
− 1 − βn − εn γ μn 1 − αn 2ξ − μn Bzn − Bp .
3.40
It follows that
2
1 − βn − εn γ μn 1 − αn 2ξ − μn Bzn − Bp
2
≤ εn γfxn − Ap xn − xn1 xn − p xn1 − p .
3.41
Bzn − Bp −→ 0.
3.42
So
16
Journal of Applied Mathematics
On the other hand, we also know that
yn − p2 PC I − μn B zn − PC I − μn B p2
≤ I − μn B zn − I − μn B p, yn − p
≤
1 I − μn B zn − I − μn B p2 yn − p2
2
2 − I − μn B zn − I − μn B p − yn − p 3.43
1 xn − p2 yn − p2 − zn − yn 2 − μn Bzn − Bp2
2
2μn zn − yn , Bzn − Bp ,
and hence
yn − p2 ≤ xn − p2 − zn − yn 2 2μn zn − yn 2 Bzn − Bp.
3.44
So
Kn − p2 αn xn 1 − αn Wn yn − p2
2
2 2 ≤ αn xn − p 1 − αn xn − p − zn − yn 2μn zn − yn Bzn − Bp
2
2
xn − p − 1 − αn zn − yn 21 − αn μn zn − yn Bzn − Bp,
xn1 − p2 εn γfxn βn xn 1 − βn I − εn A Kn − p2
2 2
2
≤ εn γfxn − Ap βn xn − p 1 − βn − εn γ Kn − p
2 2 2
≤ εn γfxn − Ap xn − p − 1 − βn − εn γ 1 − αn zn − yn 21 − αn μn 1 − βn − εn γ zn − yn Bzn − Bp.
3.45
3.46
Hence
2
1 − βn − εn γ 1 − αn zn − yn 2
≤ εn γfxn − Ap 21 − αn μn 1 − βn − εn γ zn − yn Bzn − Bp
2 2
xn − p − xn1 − p
2
≤ εn γfxn − Ap 21 − αn μn 1 − βn − εn γ zn − yn Bzn − Bp
xn − xn1 xn − p xn1 − p .
3.47
Journal of Applied Mathematics
17
From i, 3.42, and 3.26, we know that
zn − yn −→ 0.
3.48
xn − yn −→ 0.
3.49
xn − Wxn ≤ xn − Wn yn Wn yn − Wxn ≤ xn − Wn yn supWn yn − Wxn .
3.50
From 3.37 and 3.42, we can get
On the other hand, we have
By 3.30, 3.49, and using Lemma 2.7, we have
xn − Wxn −→ 0.
3.51
Step 6. We claim that lim supn → ∞ A − γfq, q − xn ≤ 0, where q PΩ I − A γfq
is the unique solution of A − γfq, x − q ≥ 0, for all x ∈ Ω.
Indeed, take a subsequence {xnj } of {xn } such that
lim sup A − γf q, q − xn lim sup A − γf q, q − xnj .
n→∞
n→∞
3.52
Since {xnj } is bounded, there exists a subsequence {xnjk } of {xnj }, which converges weakly to
p; without loss of generality, we can assume xnj p and Wxnj p, we arrive at
lim sup
n→∞
A − γf q, q − xnj
A − γf q, q − p ≤ 0.
3.53
Step 7. We show that xn → q.
Since
A − γf q, q − xn1 A − γf q, xn − xn1 A − γf q, q − xn
≤ A − γf q · xn − xn1 A − γf q, q − xn ,
3.54
so
lim sup A − γf q, q − xn1 ≤ 0.
n→∞
3.55
18
Journal of Applied Mathematics
Note that
xn1 − q2 εn γfxn βn xn 1 − βn I − εn A Kn − q2
2
εn γfxn − Aq βn xn − q 1 − βn I − εn A Kn − q 2
1 − βn I − εn A ≤ 1 − βn
Kn − q βn xn − q 1 − βn
2εn A − γf q, xn1 − q
2
2
1 − βn I − εn A ≤ 1 − βn Kn − q βn xn − q 2εn γα
1 − βn
× fxn − f q , xn1 − q 2εn γf q − Aq, xn1 − q
2
2
1 − βn I − εn A ≤ 1 − βn Kn − q βn xn − q 2εn γα
1 − βn
× xn − q · xn1 − q 2εn γf q − Aq, xn1 − q
1 − βn I − εn A2 Kn − q2 βn xn − q2 εn γα
≤
1 − βn
2 2 × xn − q xn1 − q 2εn γf q − Aq, xn1 − q
3.56
2
2
2
1 − βn − γεn
βn εn γα xn − q εn γαxn1 − q
1 − βn
≤
2εn γfxn − Aq, xn1 − q ,
which implies that
xn1 − q2 ≤
2 γ − αγ εn xn − q2
1−
1 − αγεn
2
2
γ εn2 εn
xn − q 2εn γfxn − Aq, xn1 − q .
1 − αγεn 1 − βn
3.57
Let
εn
σn 1 − αγεn
2
γ 2 εn2 xn − q 2εn γfxn − Aq, xn1 − q ,
1 − βn
2 γ − αγ εn
ρn ,
1 − αγεn
3.58
Journal of Applied Mathematics
19
then we have
lim sup
n→∞
σn
≤ 0.
ρn
3.59
Applying Lemma 2.2, we can conclude that {xn } converges strongly to q in norm. This
completes the proof.
As direct consequences of Theorem 3.1, we obtain the following corollary.
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be
a bifunction from C × C → R satisfying (A1)–(A4). Let Si : C → C be a family κi -strict
∅. Let f be a
pseudocontractions for some 0 ≤ κi < 1. Assume the set Ω ∩∞
i1 FSi ∩ EP F /
contraction of H into itself with α ∈ 0, 1 and let A be an α-inverse strongly monotone mapping. Let
F be a strongly positive linear-bounded operator on H with coefficient γ > 0 and 0 < γ < γ/α and
τ < 1. Let Wn be the mapping generated by Si and ti , where Si : C → C is a nonexpansive mapping
with a fixed point. Let {xn } and {un } be sequences generated by the following algorithm:
1
F zn , y y − zn , zn − xn ≥ 0,
λn
xn1
Kn αn xn 1 − αn Wn zn ,
εn γfxn βn xn 1 − βn I − εn A Kn ,
3.60
where {εn }, {βn }, {αn }, and {λn } are sequences in 0, 1. Assume that the control sequences satisfy
the following restrictions:
i limn → ∞ εn 0 and Σ∞
n1 εn ∞;
ii 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1;
iii 0 < lim infn → ∞ λn ≤ lim supn → ∞ λn < 1;
iv limn → ∞ |λn1 − λn | limn → ∞ |αn1 − αn | 0;
v 0 < tn ≤ b < 1;
vi λn < 2α.
Then {xn } converges strongly to w ∈ Ω where w PΩ I − A γfw.
Acknowledgments
This research is supported by the Science Research Foundation Program in the Civil Aviation
University of China 07kys08 and the Fundamental Research Funds for the Science of the
Central Universities program no. ZXH2012K001.
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